Gradient Sequence

This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.
1. Activity: Partial Derivatives from a Contour Map
In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
3. Activity: The Hill
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
4. Activity: Acting Out the Gradient
Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".
5. Activity: The Hillside
Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
6. Activity: Visualising the Gradient
Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.
7. Activity: Directional Derivatives
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.